1. Field of the Invention
The present invention relates to the design of an M-path, polyphase IIR decimate-by-M filter.
2. Description of the Related Art
Digital M-path, polyphase infinite impulse response (IIR) filters may be used as decimators (interpolators) with a decimation (interpolation) ratio M, for M greater than one. Such filters are based on the M-tap finite impulse response (FIR) filter, in which the coefficients are replaced by allpass filters. Polyphase IIR filters provide high stop band attenuation and low pass band ripple, even with a relatively small number of coefficients.
FIG. 1 shows an example of an M-path, polyphase IIR decimate-by-M filter structure 100. As shown in FIG. 1, filter 100 receives input samples x(n), which are provided to an M-1 stage tapped delay line. Input sample x(n) and the output samples of the M-1 delay line taps are down-sampled by a factor of M (steps 101-0 to 101-(m-1)). Each of the M down-sampled values are provided as input to a corresponding one of M allpass filters 102-0 to 102-(m-1). The filtered output values are summed at summer 103; thereafter, the sum is scaled by 1/M to form the output sample y(m). The transfer function of M-path, polyphase IIR decimator 100 is given by:
      H    ⁡          (      z      )        =            1      M        ⁢                  ∑                  k          =          0                M            ⁢                        z                      -            k                          ⁢                              A            k                    ⁡                      (                          z              M                        )                              Assuming that each of the M allpass filters 102-0 to 102-(M-1) has N real sections (i.e. N coefficients), the transfer functions of the allpass filters 102-0 to 102-(M-1) have the form:
            A      k        ⁡          (      z      )        =            ∏              j        =        1            N        ⁢                  ⁢                            β                      k            ,            j                          +                  z                      -            1                                      1        +                              β                          k              ,              j                                ⁢                      z                          -              1                                          Substituting the allpass filter transfer functions into the transfer function of M-path, polyphase IIR decimator 100 then gives:
      H    ⁡          (      z      )        =            1      M        ⁢                  ∑                  k          =          0                M            ⁢                        z                      -            k                          ⁢                              ∏                          j              =              1                        N                    ⁢                                                    β                                  k                  ,                  j                                            +                              z                                  -                  M                                                                    1              +                                                β                                      k                    ,                    j                                                  ⁢                                  z                                      -                    M                                                                                          Thus, the total number of filter coefficients is M×N.